Bounded Linear Operators on Ultrametric Hilbert Spaces

Bounded Linear Operators on Ultrametric Hilbert Spaces

Abstract

Let $K$ be a complete ultrametric valued field. Let $ \omega = (\omega_i)_{i\in I} \subset K\setminus \{0\}$ and let $E_\omega$ be the ultrametric Hilbert space associated with $ \omega$, i.e. the space of families $x = (x_i)_{i\in I} \subset K$ such that $\displaystyle \lim_{i\in I} x_i^2\omega_i = 0$, the limit is with respect to the Fr\'echet filter on $I$. The structure of such Hilbert spaces is far to be as for the classical Hilbert spaces. Nevertheless as a free Banach space the bounded linear operators on $E_\omega$ have a matricial description. Not any bounded linear operator has an adjoint. Some year ago we have given the charaterization of operators which admits an adjoint. These operators form a closed unitary subalgebra of the Banach algebra $\mathcal{L}(E_\omega) $ of the bounded linear operators on $E_\omega$. We shall give here some useful examples of closed subpaces of $\mathcal{L}(E_\omega).$